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The problem asks to express the absolute value function \( f(x) = |3x + 4| \) as a piecewise-defined function with linear parts.

To break down an absolute value function into a piecewise function, we consider the two cases based on when the expression inside the absolute value is positive or negative. 

### Step 1: Set up the critical point
The absolute value function changes at the point where the expression inside the absolute value equals zero:
\[
3x + 4 = 0 \quad \Rightarrow \quad x = -\frac{4}{3}
\]
This critical point divides the function into two intervals:
1. When \( x \geq -\frac{4}{3} \), the expression inside the absolute value is non-negative, so \( |3x + 4| = 3x + 4 \).
2. When \( x < -\frac{4}{3} \), the expression inside the absolute value is negative, so \( |3x + 4| = -(3x + 4) = -3x - 4 \).

### Step 2: Write the piecewise function
Thus, the piecewise-defined function is:
\[
f(x) =
\begin{cases} 
-3x - 4 & \text{if } x < -\frac{4}{3}, \\
3x + 4 & \text{if } x \geq -\frac{4}{3}.
\end{cases}
\]

This is the piecewise form of the absolute value function.

Would you like more details or have any questions?

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Here are 5 questions to expand your understanding:
1. What does the absolute value represent geometrically on a graph?
2. How do we determine the critical point for an absolute value function?
3. Why does the piecewise function have two different expressions for different values of \(x\)?
4. How would you graph the piecewise function to see the transition at \(x = -\frac{4}{3}\)?
5. Can we generalize this process to any linear absolute value function?

**Tip:** When working with absolute value functions, always focus on the point where the expression inside the absolute value equals zero to identify where the function changes behavior.

Write the absolute value function f(x) = |3x + 4| as a piecewise-defined function with linear parts.

Learn how to express the absolute value function f(x) = |3x + 4| as a piecewise-defined function. This problem explores absolute value, piecewise functions, and algebraic concepts suitable for students in Grades 9-12.

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